... J
n
q,d
is the all one matrix and I
n
q,d
is the
identity matrix, both of size n
q,d
×n
q,d
). Therefore, the largest eigenvalue of V is D
q,d
and
the absolute values of all other eigenvalues are either
(q ... over a finite field is large enough, then it contains many k-tuples of mutually
orthogonal vectors. In this note, we provide a graph theoretic proof of this resu...
... +1)
6
49n
6
.
7. The number of ascendants of a given node in a LBST
As in the case of the number of ascendants in a random BST, computing the probability that
the j
th
node in a random LBST has m ascendants ... solve the differential equations
that we had.
The last step, that of extracting coefficients in exact form, was, at large, the least system...
... 6-cycle as in the
previous case.
Remark: The proof shows that the number of Hamilton cycles is within a poly-
nomial factor of the number of two factors of G. Therefore one can generate a (near)
random ... segment containing x, the segment containing x
, and the remaining segment.
Going round the cycle C
i
, starting at x
and ending at x, the vertices x,...
... =2
[Bl]. In the general case, Bender applied a theorem of Hardy, Littlewood and Karamata
to the exponential generating function of these permutations to obtain an asymptotic
equivalent of the partial ... (single-valued) function in any simply connected domain
that avoids its singularities. The lemma shows that the singularity of C
m
at 1 determines
alone the asymptot...
... points of the planar integer lattice Z
2
,
such that for all i<jthe point A
i
is (weakly) south-east of the point A
j
, and the point
E
i
is (weakly) south-east of the point E
j
. Then the number ... 2.3 to obtain a determinant for the number of rhombus
tilings. The number of paths from a starting point A
i
to an end point, which forms an
entry in...
... n for all i.
Let a
j
be the number of rows of R that have exactly j ones. Write a = (a
0
, ,a
n
)=
a( x)inN
n+1
.Wenotethat |a| = m,andwrite
m
a
for the multinomial coefficient
m!
a
0
!·· a
n
!
With ... notation, we have the following lemma.
Lemma 6 Let a in N
n+1
satisfy |a| = m. Then the number of x in N
m
such that a( x) =a
is
m
a
.
Let λ =(λ
1...
... the drawing
of the c-graph other than their endpoints. Edges e
1
and e
2
of a c-graph G are parallel if
they cross the same edges of G, respectively. That the relation ‘parallel’ is an equivalence
relation ... edge, a representative of the set of parallel edges of the
e-graph in T . Since, according to Theorem 1, there can be at most two parallel e-graphs,
in T...
... planar
map
of G in the Euclidean plane. In particular, M has a designated outer (unbounded) face.
We denote the sets of vertices, edges and faces of a given planar map by V , E, and F,
and their ... encode the 3-coloring on the faces of the central part of
ˆ
G
k,
as
a sparse sequence a, where a
i
represents the ith square on the path P indicated by the...
... consisting of any number of
2
s and 1
s, and its subset of bipartite graphs, we characterize the optimal graphs
who maximize and minimize the number of m-matchings.
We find the expected value of ... m.
In the expression for φ(G, 4) the number of 4-cycles appeared as the first structure in
the graph, apart from n and r, which a ects the number of ma...